Faα-Irresolute Mappings

Transcription

1 BULLETIN o he Bull. Malaysian Mah. Sc. Soc. (Second Series) 24 (2001) MLYSIN MTHEMTICL SCIENCES SOCIETY Faα-Irresolue Mappings 1 R.K. SRF, 2 M. CLDS ND 3 SEEM MISHR 1 Deparmen o Mahemaics, Governmen K.N.G. College Damoh (M.P.) India 2 Universidade Federal Fluminense, Deparemeno Maemaica plicada Rua Mario Sanos Braga, s/n Nierio CEP: , R.J. Brasil 3 Deparmen o Mahemaics, Governmen uonomous Science College, Jabalpur (M.P.) India bsrac. The presen noe inroduces a new class o mappings called Faα-irresolue mappings. We obain several characerizaions o his class and sudy is properies and invesigae he relaionship wih he known mappings. 1. Inroducion T.H. Yalvac [10] inroduced he noion o uzzy irresolue mappings. The presen auhor [9] inroduced he noion o uzzy semi α-irresolue (shorly Fsα-irresolue) mappings. The purpose o his noe is o inroduce and invesigae he concep o Faα-irresolue mappings and give several characerizaion and is properies. Relaion beween his class and oher classes o uncions are obained. The class o uzzy almos α-irresolue mapping, which is sronger han Fβ-coninuiy, is a generalizaion o boh Fa-irresolue mappings and Fsα-irresolue mappings. 2. Preliminaries Throughou his noe, spaces always mean uzzy opological spaces and : X Y denoes a mapping rom a space ( X, τ ) ino a space ( Y, σ ). The closure and he inerior o are denoed by C 1( ) and ), respecively. Deiniion 2.1. uzzy subse o space X is said o be uzzy α-open [3], (resp. uzzy semi open [1], uzzy pre open [3], uzzy β-open [4]) i (resp. )), )), ). The amily o all Fα-open (resp. Fs-open Fp-open, Fβ-open) ses in a space X is denoed by Fα O(X ) (resp. FSO ( X ), FPO( X ) and Fβ O(X )). The complemen o a Fα-open (resp. Fβ-open) se is said o be Fα-closed (resp. Fβ-closed). The inersecion

2 194 R.K. Sara e al. o all Fα-closed ses conaining is called α-closure o and is denoed by α ); he union o all Fα-open se conained in is called he α-inerior o and is denoed by α ). [6] Deiniion 2.2. mapping : X Y is said o be uzzy α-irresolue [6] (resp. uzzy semi α-irresolue [9] ) i ( ) is Fα-open (resp. Fs-open) in X or every Fα-open se o Y. Deiniion 2.3. mapping : X Y is said o be uzzy srongly α-coninuous [9] (resp. uzzy irresolue [10], Fa-irresolue [8] ) i ( ) is Fα-open (resp. Fs-open, Fβ-open) in X or every Fs-open se o Y. Deiniion 2.4. mapping : X Y is said o be Fβ-coninuous [4] (resp. Fβ-irresolue) i ( ) o Y. is Fβ-open in X or every uzzy open (resp. Fβ-open) se Deiniion 2.5. uzzy poin x is said o be quasi-coinciden wih a uzzy se in X i + ( > 1. uzzy se in X is said o be quasi- coinciden wih a uzzy se B in X, denoed by q B, i here exiss a poin x in X, such ha, ( + B( > 1. [7] Lemma 2.1. Le : X Y be a mapping and x be a uzzy poin o X. Then, (a) ( x ) 1 q B x q ( B), or every uzzy se B o Y. (b) x q ( x ) q ( ), or every uzzy se o X. [10] 3. Faαirresolue mappings Deiniion 3.1. mapping : X Y is said o be Faα-irresolue i ( ) β-open (shorly Fβ-open) in X or every Fα-open se o Y. is uzzy From he deiniion, we obain he ollowing diagram: Fα-irresolue Fsα-irresolue Faα -irresolue Fβ-coninuiy uzzy srongly α-coninuiy uzzy irresolue Fa-irresolue Fβ-irresolue The examples given below shows ha he converse o hese implicaions are no rue in general. For,

3 Faα-Irresolue Mappings 195 Example 3.1. Le X = { x, y}, Y = { a, b}. The uzzy se, B, H, E, G are deined as: ( a) = 0.5, = ( b) = 0.6 ; B( = 0, B ( y) = 0.6; H ( a) = 0, H ( b) = 0.8; E( = 0.6, E ( y) = 0.3; G ( a) = 0.2, G( b) = 0.6. Le τ = { 0,1}, σ = {0,,1}. τ = {0,,1}, 1 B σ = {0,,1}, τ = {0,,1}, σ = {0,,1}. Then he mapping : ( x, τ ) ( Y, σ ) 1 H deined by 2 E 2 g ( = a and ( y) = b is Fβ-irresolue and hence Faα-irresolue bu no Fsα-irresolue; he mapping g ( X, τ ) ( Y, ) deined by ( = a, ( y) = b is : 1 σ 1 Fα-irresolue bu no Fa-irresolue. The mapping h: X, τ ) ( Y, ) deined by ( = a, ( y) = b is Fβ-coninuous bu no Faα-coninuous. ( 2 σ 2 Example 3.2. Le X = { x, y}, Y = { a, b}, a uzzy ses is deined as: ( = 0.5, ( y) = 0.3. Le τ = { 0,, 1} and σ = {0,1} hen he mapping : ( X, τ ) ( Y, σ ) deined by ( = a, ( y) = b is uzzy srongly α-coninuous bu no Fβ-irresolue. Oher examples can be seen in [6]. Theorem 3.1. The ollowing are equivalen or a mapping : X Y, (a) (b) (c) (d) (e) is Faα-irresolue; or every uzzy poin x in X and every Fα-open se V o Y conaining ( x ), here exiss an Fβ-open se U o X conaining x such ha ( U ) V ; or every uzzy poin x o X and or every Fα-open se V o Y conaining ( x ), here exiss an Fβ-open se U in X such ha x 1 U ( V ); or every uzzy poin x in X, he inverse image o each α-neighbourhood [7] o ( x ) is a β-neighbourhood [6] o x. or every uzzy poin x in X and each α-neighbourhood B o ( x ) here exiss a β-neighbourhood o x such ha ( ) B ; 1 1 () ( V ) In ( ( V )))), or every Fα-open se V o Y; 1 H (g) ( ) is Fβ-closed in X, or every Fα-closed se H o Y; 1 1 (h) In ( In ( ( ) ( α or every uzzy subse B o Y; (i) ( In ( α ( )) or every uzzy subse o X. Proo. (a) (b) (c); (d) (e): obvious (b) (): Le V be any Fα-open se o Y and x 1 ( V ). By (b), here exiss a Fβ-open se U o X conaining x such ha ( U ) V. Thus we have x U U ) ) ) In ( 1 ( V ) ) ) and hence ( V ) In ( 1 ( V )) ) ).

4 196 R.K. Sara e al. () (g): Le H be any Fα-closed se o Y. Se V = Y H, hen V is 1 Fα-open in Y. By (), we obain ( V ) ( V ) ) ) ) and hence, 1 1 ( H ) = X ( Y H ) = X ( V ) is Fβ-closed in X. 1 (g) (h): Le B be any uzzy sub se o Y. Since α B) is an Fα-closed subse o 1 Y, ( α( is Fβ-closed in X, and hence ( αc1)( ))) ( α. Thereore, we obain ( )) ( α. 1 (h) (i): Le be any uzzy subse o X by (h), we have InC 1( 1 1 ( ( )) ( α ( )) and hence ( ) α ( )). (i) (a): Le V be any Fα-open subse o Y. Since ( Y V ) 1 V = X ( ) is a uzzy subse o X, by (), we obain ( 1 X ( Y V )) ) ) ) α ( ( Y V ))) = Y α V ) = Y V, and hence, 1 ( V )))) 1 = ( In ( X ( V )))) 1 = ( In ( In ( Y V )))))) ( ( ( Y V )))))) ( Y V ) = X ( V ). Thereore, we 1 1 V have ( V ) ( V )))) and hence ( ) is Fβ-open in X. Thus is Faα-irresolue. (a) (d): Le x be a uzzy poin in X and V be α-neighbourhood o ( x ), hen 1 G here exis a Fα-open se G o, Y such ha, ( x ) G V. Now ( ) is Fβ-open 1 1 in X and ( G) ( V ). Thus ( ) x V is a β-neighbourhood o x in X. (e) (b): Le x be a uzzy poin in X and V is Fα-open se o Y such ha ( x ) V. Then V is α-neighbourhood o x ), so here is a β-neighbourhood o x, such ha x, and ( ) V. Hence here exiss a Fβ-open se U in X such ha U and so ( U ) ( ) V. x Theorem 3.2. The ollowing are equivalen or a mapping (a) is Faα-irresolue ; (b) (c) ( : X Y : or each poin x o X and every Fα-open se B o Y, such ha ( x ) q B, here exiss, a Fβ-open se in X such ha x q and ( ) B. For every uzzy poin x o X and every Fα-open se B o Y such ha ( x ) q B, here exiss, a Fβ-open se o X such ha 1 B x q and ( ).

5 Proo. (a) (b): Le 1 B Faα-Irresolue Mappings 197 x be a uzzy poin o X and B be a Fα-open se o Y such ha 1 B ( x ) q B. Then ( ) is Fβ-open in X, and q ( ) by Lemma 2.1. I we ake 1 B = ( ) hen x q and ( ) = ( ( B. (b) (c): Le x be a uzzy poin and B be a Fα-open se o Y such ha ( x ) q B. Then by (b), here exiss a Fβ-open se o X such ha x q and ( ) B. Hence we have (c) (a): x x q and ( ( )) ( B). 1 B x Le B be a Fα-open se o Y and x be a uzy poin o X such ha ( ). Then ( x ) B. Choose he uzzy poin x ( = 1 x (. Then c ( x ) q B. nd so by (c), here exiss a Fβ-open se o X such ha x c q and c ( ) B. Now x c q implies x ( + ( = 1 x ( + ( > 1. I ollows 1 B ha x. Thus ( ). Hence ( ) is Fβ-open in X. x Theorem 3.3. mapping 1 B : X Y is Faα-irresolue i he graph mapping g : X X Y, deined by g ( = ( x, ( ) or each x X, is Faα-irresolue. Proo. Le x X and V be any Fα-open se o Y conaining ( x ). Then, X V is Fα-open is X Y conaining g ( x ). Since g is Faα-irresolue here exiss, Fβ-open se U o X conaining x such ha g ( U ) X V and hence ( U ) V. Thus, is Fa α -irresolue. Theorem 3.4. I : X Y is Faα-irresolue and is Fα-open subse o X, hen he resricion : Y is Faα-irresolue. Proo. Le V be any Fα-open se o Y. Since is Fa α -irresolue, hen ( ) is 1 1 Fβ-open in X. Since is Fα-open in X, hen ( ) ( V ) = ( V ) is Fβ-open in and hence is Faα-irresolue. c V Theorem 3.5. Le : X Y be a mapping and { i : i ^} be a cover o X by Fβ-open ses o X. Then is Faα-irresolue i i : i Y is Faα-irresolue or each i ^. Proo. Le V be any Fα-open se o Y. Since i is Faα-irresolue, hen ( ) 1 ( V ) is Fβ-open in and since FβO(X ), hen ( ) 1 ( V ) is Fβ-open i in X or each 1 i i i 1 1 ( V ) = X ( V ) = { i i ^. Thereore ( V ) : i ^} = { i ) ( V ) : i ^} is Fβ-open in X. Hence is Faα-irresolue. 1

6 198 R.K. Sara e al. B Theorem 3.6. mapping : X Y is Faα-irresolue, hen ( ) X or any nowhere dense se B o Y. is Fβ-closed in Proo. Le B be any nowhere dense subse o Y, hen Y B is Fα-open in Y. Since is 1 1 Faα-irresolue, hen ( Y B) = X ( B) is Fβ-open in X and hence ( B) is Fβ-closed in X. Theorem 3.7. mapping : X Y is Faα-irresolue i, or each p Y and each uzzy open se V o Y such ha p Y and each uzzy open se V o Y such ha p V )), he inverse image o V p is Fβ-open in X. Proo. Necessiy. Since V V p V )), hen V p is a Fα-open se o Y. Since is Faα-irresolue, hen ( V p) is Fβ-open in X. 1 Suiciency. Since V be a Fα-open se o Y. Then, here exiss an uzzy open se B o 1 Y such ha B V <. By hypohesis, ( B p) is Fβ-open in X or 1 each p V. This shows ha ( V ) = { ( B p) : p V} hence is Faα-irresolue. 1 is Fβ-open in X and Theorem 3.8. Le : X Y and g : Y Z be mapping. Then he composiion g o : X Z is Faα-irresolue i and g saisy one o he ollowing condiion: (a) (b) (c) is Faα-irresolue and g is Fα-irresolue, is Fβ-irresolue and g is Faα-irresolue, is Fa-irresolue and g is Fsα-irresolue. g 1 W Proo. Le W be any α-open subse o Z. Since g is Fα-irresolue, hen ( ) is Fβ-open in Y. Since is Faα-irresolue, ( g o ) ( W ) = ( g ( W )) is Fβ-open in X and hence g o is Faα-irresolue. The proo o he condiions (b) and (c) is analogous o ha o (a). I ollows rom he deiniions. We recall ha a space X is said o be uzzy submaximal i every uzzy dense subse o X is uzzy open in X and uzzy exremely disconneced i he closure o each uzzy open se o X is uzzy open in X. Theorem 3.9. Le X be a uzzy submaximal and uzzy exremely disconneced space. Then he ollowing are equivalen or a mapping : X Y : (a) is Fα-irresolue; (b) is Fsα-irresolue; (c) is Faα-irresolue;

7 Faα-Irresolue Mappings 199 Proo. This ollows rom he ac ha i ( X, τ ) is uzzy submaximal and exremely disconneced hen τ = FαO( X ) = FSO( X ) = FβO( X ). cknowledgemen. The auhors wish o express heir sincere graiude o reeree or his valuable suggesions o improve he qualiy o his paper. Reerences 1. K.K. zad, On uzzy semiconinuiy, uzzy almos coninuiy and uzzy weakly coninuiy, J. Mah. nal. ppl. 82 (1981), Y. Beceren, lmos α-irresolue mappings; Bull. Cal. Mah. Soc. 92 (2000), S. Bin Shahna, On uzzy srongly semiconinuiy and uzzy preconinuiy, Fuzzy se and sysems 14 (1991), S. Mashhour, M.H. Ghanim and Fah lla, On uzzy nonconinuous mapings, Bull. Cal. Mah. Soc. 78 (1986), P.P. Ming and L.Y. Ming, Fuzzy opology II produc and quoien space, J. Mah. nal. ppl. 77 (1980), R. Prasad, S.S. Thakur and R.K. Sara, Fuzzy α-irresolue mapping, Jour. Fuzzy Mah. 2 (1994), P.M. Pu and Y.M. Liu, Fuzzy opology I, neighbourhood srucure o a uzzy poin and Moore smih convergence, J. Mah nal. ppl. 76 (1989), R.K. Sara, M. Caldas and Mishra Seema, Fa-irresolue mappings (Under Preparaion). 9. R.K. Sara, M. Caldas and Mishra Seema, Fuzzy semi α-irresolue mapping (submied). 10. T.H. Yalvac, Fuzzy se and uncion on uzzy space, J. Mah. nal. ppl. 126 (1987), L.. Zadeh, Fuzzy ses, inorm and conrol 8 (1965), Keyword: uzzy irresolue, Fα-open se, Fβ-open se, Fα-irresolue mapping Mahemaics Subjec Classiicaion: 5440